Quantized Fourier and Polynomial Features for more Expressive Tensor Network Models

In the context of kernel machines, polynomial and Fourier features are commonly used to provide a nonlinear extension to linear models by mapping the data to a higher-dimensional space. Unless one considers the dual formulation of the learning problem, which renders exact large-scale learning unfeasible, the exponential increase of model parameters in the dimensionality of the data caused by their tensor-product structure prohibits to tackle high-dimensional problems. One of the possible approaches to circumvent this exponential scaling is to exploit the tensor structure present in the features by constraining the model weights to be an underparametrized tensor network. In this paper we quantize, i.e. further tensorize, polynomial and Fourier features. Based on this feature quantization we propose to quantize the associated model weights, yielding quantized models. We show that, for the same number of model parameters, the resulting quantized models have a higher bound on the VC-dimension as opposed to their non-quantized counterparts, at no additional computational cost while learning from identical features. We verify experimentally how this additional tensorization regularizes the learning problem by prioritizing the most salient features in the data and how it provides models with increased generalization capabilities. We finally benchmark our approach on large regression task, achieving state-of-the-art results on a laptop computer

Sparse Gaussian Processes in the Longstaff-Schwartz algorithm

In financial applications it is often necessary to determine conditional expectations in Monte Carlo type of simulations. The industry standard at the moment relies on linear regression, which is characterized by the inconvenient problem of having to choose the type and number of basis functions used to build the model, task which is made harder by the frequent impossibility to use an alternative numerical method to evaluate the "ground truth". In this thesis Gaussian Process Regression is investigated as potential substitute for linear regression, as it is a flexible Bayesian non-parametric regression model, which requires little tuning to be used. Its downfall is the computational complexity related to its "training" phase, namely cubic, which requires the use of algorithmic approximations. The most prominent approximations are reviewed and tested in different scenarios requiring the approximationof conditional expectation by regression, among which the Longstaff-Schwartz algorithm for the pricing of Bermudan options. This thesis was carried out in cooperation with ABN-AMRO bank

Towards Green AI with tensor networks -- Sustainability and innovation enabled by efficient algorithms

The current standard to compare the performance of AI algorithms is mainly based on one criterion: the model's accuracy. In this context, algorithms with a higher accuracy (or similar measures) are considered as better. To achieve new state-of-the-art results, algorithmic development is accompanied by an exponentially increasing amount of compute. While this has enabled AI research to achieve remarkable results, AI progress comes at a cost: it is unsustainable. In this paper, we present a promising tool for sustainable and thus Green AI: tensor networks (TNs). Being an established tool from multilinear algebra, TNs have the capability to improve efficiency without compromising accuracy. Since they can reduce compute significantly, we would like to highlight their potential for Green AI. We elaborate in both a kernel machine and deep learning setting how efficiency gains can be achieved with TNs. Furthermore, we argue that better algorithms should be evaluated in terms of both accuracy and efficiency. To that end, we discuss different efficiency criteria and analyze efficiency in an exemplifying experimental setting for kernel ridge regression. With this paper, we want to raise awareness about Green AI and showcase its positive impact on sustainability and AI research. Our key contribution is to demonstrate that TNs enable efficient algorithms and therefore contribute towards Green AI. In this sense, TNs pave the way for better algorithms in AI